Q10.5.1
In Exercises 10.5.1-10.5.12 find the general solution.
1. y′=[34−17]y
2. y′=[0−11−2]y
3. y′=[−74−1−11]y
4. y′=[31−11]y
5. y′=[412−3−8]y
6. y′=[−109−42]y
7. y′=[−1316−911]y
8. y′=[021−461042]y
9. y′=13[11−3−4−43−210]y
10. y′=[−11−1−202−13−1]y
11. y′=[4−2−2−23−12−13]y
12. y′=[6−532−13211]y
Q10.5.2
In Exercises 10.5.13-10.5.23 solve the initial value problem.
13. y′=[−118−2−3]y,y(0)=[62]
14. y′=[15−916−9]y,y(0)=[58]
15. y′=[−3−41−7]y,y(0)=[23]
16. y′=[−724−617]y,y(0)=[31]
17. y′=[−73−3−1]y,y(0)=[02]
18. y′=[−1101−1−2−1−1−1]y,y(0)=[65−7]
19. y′=[−221−221−332]y,y(0)=[−6−20]
20. y′=[−7−44−101−9−56]y,y(0)=[−69−1]
21. y′=[−1−4−1361−3−23]y,y(0)=[−213]
22. y′=[4−8−4−3−1−31−19]y,y(0)=[−41−3]
23. y′=[−5−111−7113−408]y,y(0)=[022]
Q10.5.3
The coefficient matrices in Exercises 10.5.24-10.5.32 have eigenvalues of multiplicity 3. Find the general solution.
24. y′=[5−11−19−3−224]y
25. y′=[110−122232−16]y
26. y′=[−6−4−42−11231]y
27. y′=[02−2−15−3111]y
28. y′=[−2−12102−24112−248]y
29. y′=[−1−1281−941−61]y
30. y′=[−40−1−1−3−110−2]y
31. y′=[−3−3445−823−5]y
32. y′=[−3−101−10−1−1−2]y
Q10.5.4
33. Under the assumptions of Theorem 10.5.1, suppose u and ˆu are vectors such that
(A−λ1I)u=xand (A−λ1I)ˆu=x,
and let
y2=ueλ1t+xteλ1tand ˆy2=ˆueλ1t+xteλ1t.
Show that y2−ˆy2 is a scalar multiple of y1=xeλ1t.
34. Under the assumptions of Theorem 10.5.2, let
y1=xeλ1t,y2=ueλ1t+xteλ1t, and y3=veλ1t+uteλ1t+xt2eλ1t2.
Complete the proof of Theorem 10.5.2 by showing that y3 is a solution of y′=Ay and that {y1,y2,y3} is linearly independent.
35. Suppose the matrix A=[a11a12a21a22] has a repeated eigenvalue λ1 and the associated eigenspace is one-dimensional. Let x be a λ1-eigenvector of A. Show that if (A−λ1I)u1=x and (A−λ1I)u2=x, then u2−u1 is parallel to x. Conclude from this that all vectors u such that (A−λ1I)u=x define the same positive and negative half-planes with respect to the line L through the origin parallel to x.
Q10.5.5
In Exercises 10.5.36-10.5.45 plot trajectories of the given system.
36. y′=[−3−141]y
37. y′=[2−110]y
38. y′=[−1−335]y
39. y′=[−53−31]y
40. y′=[−2−334]y
41. y′=[−4−332]y
42. y′=[0−11−2]y
43. y′=[01−12]y
44. y′=[−21−10]y
45. y′=[0−41−4]y
